# Theoretical and Actual Cycle Analysis

Post Author : Blue Collar Post Date : 12/3/20 Post Time : December 03, 2020

The thermodynamic analysis presented here is an outline of the air-standard Brayton cycle and its various modifications. These modifications are evaluate to examine the effects they have on the basic cycle. One of the most important is the augmentation of power in a gas turbine. This is treated in a special section in this chapter. The Brayton Cycle

The Brayton cycle in its ideal form consists of two isobaric processes and two isentropic processes. The two isobaric processes consist of the combustor system of the gas turbine and the gas side of the HRSG. The two isentropic processes represent the compression (Compressor) and the expansion (Turbine Expander) processes in the gas turbine. Figure 2-1 shows the Ideal Brayton Cycle.

A simplified application of the first law of thermodynamics to the air-standar Brayton cycle in Figure 2-1 (assuming no changes in kinetic and potential energy has the following relationships: Work of compressor

Wc = ˙ma(h2 − h1) (2-1)

Work of turbine

Wt = ( ˙ma + ˙ mf )(h3 − h4) (2-2)

Total output work

Wcyc = Wt − Wc (2-3)

57

Q2,3 = ˙mf xLHVfuel = ( ˙ma + ˙ mf )(h3)− ˙ mah2 (2-4)

Thus, the overall cycle efficiency is

ηcyc = Wcyc/Q2,3 (2-5)

Increasing the pressure ratio and the turbine firing temperature increases the

Brayton cycle efficiency. This relationship of overall cycle efficiency is based on

certain simplification assumptions such as: (1) ˙ma  ˙mf , (2) the gas is calorically

and thermally perfect, which means that the specific heat at constant pressure (cp)

and the specific heat at constant volume (cv) are constant thus the specific heat

ratio γ remains constant throughout the cycle, (3) the pressure ratio (rp) in both

the compressor and the turbine are the same, and (4) all components operate at

100% efficiency. With these assumptions the effect on the ideal cycle efficiency

as a function of pressure ratio for the ideal Brayton cycle operating between

the ambient temperature and the firing temperature is given by the following

relationship:

ηideal =



1 − 1

r

γ−1

γ

p



(2-6)

where Pr = Pressure Ratio; and γ is the ratio of the specific heats. The above

equation tends to go to very high numbers as the pressure ratio is increased.

Assuming that the pressure ratio is the same in both the compressor and the

turbine the following relationships hold using the pressure ratio in the compressor:

ηideal = 1 − T1

T2

(2-7)

and using the pressure ratio in the turbine

ηideal = 1 − T4

T3

(2-8)

In the case of the actual cycle the effect of the turbine compressor (ηc) and

expander (ηt ) efficiencies must also be taken into account to obtain the overall

cycle efficiency between the firing temperature Tf and the ambient temperature

Tamb of the turbine. This relationship is given in the following equation:

ηcycle =



ηtTf − Tambr



γ−1

γ



p

ηc

Tf − Tamb − Tamb

rp



γ−1

γ



− 1

ηc





1 − 1

r



γ−1

γ



p



(2-9)

Figure 2-2 shows the effect on the overall cycle efficiency of the increasing

pressure ratio and the firing temperature. The increase in the pressure ratio

increases the overall efficiency at a given firing temperature; however, increasing

the pressure ratio beyond a certain value at any given firing temperature can

actually result in lowering the overall cycle efficiency. It should also be noted

that the very high-pressure ratios tend to reduce the operating range of the turbine

compressor. This causes the turbine compressor to be much more intolerant

to dirt build-up in the inlet air filter and on the compressor blades and creates

large drops in cycle efficiency and performance. In some cases, it can lead to

Figure 2-2. Overall cycle efficiency as a function of the firing temperature and

pressure ratio. Based on a compressor efficiency of 87% and a turbine efficiency

of 92%.

compressor surge, which in turn can lead to a flameout, or even serious damage

and failure of the compressor blades and the radial and thrust bearings of the gas

turbine.

To obtain a more accurate relationship between the overall thermal efficiency

and the inlet turbine temperatures, overall pressure ratios, and output work,

consider the following relationships. For maximum overall thermal cycle efficiency,

the following equation gives the optimum pressure ratio for fixed inlet

temperatures and efficiencies to the compressor and turbine:

(rp)copt =

1

T1T3η1 − T1T3 + T 2

1

[T1T3ηt

(T1T3ηt )2 − (T1T3ηt − T1T3 + T 2

1 )(T 2

3 ηcηt − T1T3ηcηt + T1T3ηt )]

γ

γ−1

(2-10)

The above equation for no losses in the compressor and turbine (ηc = ηt = 1)

reduces to:

(rp)copt =

T1T3

T 2

1

γ

γ−1

(2-11)

The optimum pressure ratio for maximum output work for a turbine taking

into account the efficiencies of the compressor and the turbine expander section

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